MAA  Mathematical Association of America

Seaway Section

Location: Valentine 103

Note: Student Talks

1. Time:

   1:15 pm – 1:35 pm

   Title:

   Exploring properties of reversed digit pairs

   Speaker:

   Elise Heppell (St. Lawrence University)

   Abstract

   This talk covers the results of my fellowship project at St Lawrence University this past summer, the aim of which was to identify, write formulae for and write proofs for some patterns in reversed digit pairs of numbers. We’ll primarily focus on what happens when taking the difference of fully reversed, partially reversed and scrambled digit pairs, as well as foray into changing the base of the number system and performing other operations on these pairs of numbers.

2. Time:

   1:45 pm – 2:05 pm

   Title:

   Using Isometries to Investigate Equivalence of Curves in Three Dimensions

   Speaker:

   Daniel Schlagel (SUNY Oneonta)

   Abstract

   In this presentation, we will begin by discussing isometries of the Euclidean space regarded as compositions of orthogonal transformations and translations. We will then bring vector fields into play to investigate more intricate differential geometric aspects, such as tangent maps and Frenet frame fields, in order to characterize when a curve can be mapped to another curve via an isometry.

3. Time:

   2:15 pm – 2:35 pm

   Title:

   Character Codegrees: A Solution to the Classification of Nonsolvable Groups with Square-Free GCD Degree and Codegree

   Speaker:

   Karam Aldahleh (University of Rochester)

   Abstract

   Let $G$ be a finite group and $\chi$ be an irreducible character of $G$. The codegree of $\chi$ is defined as $\chi^c(1) =\frac{|G: \ker\chi|}{\chi(1)}$. In a paper by Gao, Wang, and Chen, it was shown that $G$ cannot satisfy the condition that $\gcd(\chi(1),\chi^c(1))$ is prime for all $\chi\in\text{Irr}(G)^\#$. We generalize this theorem by solving one of Guohua Qian's unsolved problems on character codegrees. In Qian's survey article, he inquires about the structure of non-solvable finite groups with square-free $\gcd$ instead. In particular, we prove that if $G$ is such that $\gcd(\chi(1),\chi^c(1))$ is square-free for every irreducible character $\chi$, then $G/\text{Sol}(G)$ is isomorphic to one among a particular list of almost simple groups.

4. Time:

   2:45 pm – 3:05 pm

   Title:

   Frobenius Case of Qian's Conjecture and Restrictions on a Counterexample

   Speaker:

   Karam Aldahleh (University of Rochester)

   Abstract

   Gouhua Qian established in a note of his that if $G$ is a finite solvable group, then for every $g\in G$ there exists some $\chi\in\text{Irr}(G)$ where $p\;|\;\chi^c(1)$ for each prime $p$ dividing $o(g)$. Qian later generalized this result in another note, proving that for every $g\in G$ there exists some $\chi\in\text{Irr}(G)$ such that $o(g)\;|\;\chi^c(1)$. He further conjectured that arbitrary finite groups satisfy this property. Since then, Eugenio Giannelli and Sesuai Y. Madanha solved the conjecture for almost simple groups while Zeinab Akhlaghi, et al. proved it for groups with trivial Fitting subgroup. The latter result also established a useful setup for a minimality argument, where a supposed minimal counterexample must have solvable socle. In this paper, we solve the conjecture for all Frobenius groups. We then expand on the minimality setup by demonstrating a further restriction on the commutator subgroup.

Location: Valentine 105

Note: Student Talks

1. Time:

   1:15 pm – 1:35 pm

   Title:

   Learning Interactions in Collective Dynamics

   Speakers:

   Nipuni Senani De Silva Rammini (Clarkson University), James Greene (Clarkson University), Ming Zhong (University of Houston)

   Abstract

   Interacting particle systems, also known as agent-based models (ABMs), represent one category of dynamical systems that are used to study a wide range of physical phenomena across multiple scales. Examples from science and engineering include cell migration, swarm robotics, social psychology, and animal migration patterns and interactions. A ubiquitous feature of such systems is that they exhibit a form of emergence: local interactions leading to large-scale coordination. A fundamental scientific question is thus to understand the local interactions that give rise to the observed emergent dynamics. We are interested in methods for learning interactions generally, which can describe any ABM defined by an interaction kernel without making any additional assumptions about the analytical form of this kernel (i.e. it is non-parametric). The advantage of this kernel-based approach is that it incorporates the underlying physics of the model (i.e. collective dynamics), which more general approaches may ignore, potentially limiting their effectiveness. We propose to extend a non-parametric statistical learning approach for learning the interaction kernel for systems with both self-propulsion and collective dynamics, given an observed set of trajectories. First, we parametrically learn the intra-agent force while simultaneously inferring the interaction kernel non-parametrically. The method is validated on two well-known models. We extended this approach to learn the intra-agent force non-parametrically.

2. Time:

   1:45 pm – 2:05 pm

   Title:

   Quantifying the Effect of Space on Antibiotic Resistance Evolution.

   Speaker:

   Induni Kariyawasam Mg (Clarkson University)

   Abstract

   Antibiotics, which can be defined as substances that work against bacteria, are one of the most useful agents used in healthcare. As a result, they serve to treat and prevent many bacterial infections. Since bacteria live in diverse spatially structured environments that can impact their evolutionary dynamics, it is important to study the effect of space on evolution in presence and absence of antibiotics. According to the experimental setup, bacteria are growing under a limited nutrient supply, both in the presence and absence of antibiotics. As nutrients in the culture medium are depleting due to bacterial consumption, the bacteria face intense selective pressure to adapt. This lack of resources leads to changes in the bacteria, including genetic mutations that can enhance survival strategies such as increased motility. Bacteria that are randomly selected or that undergo beneficial mutations for motility move toward nutrients to survive. Since limited nutrients affect population density, nutrient consumption becomes a factor in altering the division rate of the system. As a result, the division rate is influenced not only by the system’s current state but also by its past dynamics. This brings the Non-Markovian nature to the process where division rate of an individual is a function time as well as state. As an initial step in this project—prior to studying changes in the system as a function of spatial diversity and the presence or absence of antibiotics under limited nutrient supply—we are analyzing the statistics of a time- and state-dependent non-Markovian process.

3. Time:

   2:15 pm – 2:35 pm

   Title:

   Machine Learning for Estimating Wind Patterns from Grass Orientation

   Speaker:

   Francesca Mnenula (St. Lawrence University)

   Abstract

   We explore the use of Histogram of Oriented Gradients (HOG) to estimate grass orientation from images generated by generative artificial intelligence models. These grass orientations computed with HOG are compared to the wind direction and the grass orientation specified in the AI model's prompt. This preliminary simulation evaluates the HOG method’s ability to recover grass orientation under controlled conditions. These results provide a first step toward potential applications that could help communities monitor changes in local wind patterns from grass images in fields amidst climate-driven environmental changes.

4. Time:

   2:45 pm – 3:05 pm

   Title:

   A meshfree RBF–ETD scheme for solving high-dimensional Black–Scholes Problems with non-smooth payoffs

   Speakers:

   Ibraheem Abiodun Yahayah (Clarkson University), Guangming Yao (Clarkson University), Emmanuel Asante-Asamani (Clarkson University)

   Abstract

   The Black-Scholes equation is a bedrock in financial mathematics, widely used for pricing options and derivatives. However, its application faces major challenges, like non-smooth initial conditions, high dimensionality in multi-asset scenarios, and nonlinearities arising from factors like transaction costs or volatility dependent on option price gradients. To address these complexities, we propose an RBF-ETD scheme that combines Radial Basis Functions (RBF) for spatial approximation with Exponential Time Differencing (ETD) for efficient time integration.This approach leverages the flexibility of RBFs in solving high-dimensional PDEs and ETD’s ability to handle nonlinearities and non-smooth data without iterative solvers. We validate the method by comparing it with established schemes such as Crank Nicolson with RBF (CN-RBF) to examine its comparative accuracy, stability, and computational efficiency. The proposed RBF-ETD scheme promises a robust tool for solving complex Black-Scholes models in real-world financial applications.